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182 Chapter 8
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Microcirculation
model
Model of
tumor growth
Figure 8.5
Block scheme of coupling of models. NA
angiogenesis; NP
, NAare the numbers of degraded normal and angiogenic capillaries; PC, AC
þ
is the number of capillaries appeared in process of
are the surface densities of normal and angiogenic capillaries, Q is the blood flow density.
submodels occurs at a constant time interval determined by the rate of microcirculatory
network restructuring. This interval is chosen in such a way that, in each calculated layer
of the microcirculatory network model, the number of capillaries that appear or degrade
during this interval is no more than a hundred. This condition ensures a smooth change in
the structure of the capillary network and thus a smooth change in the inflow of nutrients.
The basic scheme of interactions between submodels is shown in Fig. 8.5.
The submodel of tumor growth provides information on the number and location in space
of new angiogenic capillaries NA
þ
that arise in the submodel of the microcirculatory
network. The tumor submodel also informs the whole model about degraded preexisting
and angiogenic capillaries NP
Since we do not consider antiangiogenic therapy yet, there is no term NP
and NA, these capillaries are removed from the graphs.
þ
corresponding
to the process of capillary normalization. Information about new capillaries is determined
by the distribution of VEGF, denoted by variable V in tumor model; information about
destroyed ones is determined by the interaction of the tumor (fraction of cells along with
necrosis in tissue is denoted by variable n
), with the microcirculatory network, which is
t
described in the tumor growth model by two variables of normal and angiogenic capillary
surface density PC and AC. Each transferred value refers to a specific i-th spherical layer
and j-th time interval of the microcirculatory network model. The corresponding equations
are as follows:
NA
NP
NA
þ
i;j
i;j
i;j
¼ 4p
¼ 4p
¼ 4p
Z
Z
t
r
j
i
t
r
j1
i1
Z
Z
t
r
j
i
Vðr; tÞ
R
Vðr; tÞþV
ðPCjðrÞþACjðrÞÞ 1
PCjðrÞþACjðrÞ
C
max
½kntðr; tÞPCjðrÞr2drdt;
t
r
j1
i1
Z
Z
t
r
j
i
½kntðr; tÞACjðrÞr2drdt;
t
r
j1
i1
r
2
drdt;
(8.13)
where R is the maximum rate of tumor angiogenesis, which is achieved under levels of
VEGF, sufficiently greater than V
density; and k determines the rate of capillary degradation under the influence of
; C
is the maximum possible capillary surface
max
Hemodynamics in normal and angiogenic capillary networks 183
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malignant cells and necrotic tissue. Note that the influence of VEGF on the structure and
functionality of normal capillaries is yet not accounted for.
At each moment of the data exchange t
, on the basis of this data, the structure of the
j
microvascular network is updated. Angiogenic capillaries are formed as bridges between
two already existing vascular elements. The distance between these elements does not
exceed 1 mm. The lengths and diameters of angiogenic capillaries are assigned to
correspond the known experimental statistical distribution, discussed in Section 8.2.2.
Parameter k
> 1 is assigned to angiogenic capillaries. It influences their resistance and
uv
allows to account for their tortuosity and compression (see Eqs. 8.11 and 8.12). If a
degraded capillary is the only one that supplies a set of other capillaries, they are
destroyed as well. However, the number of these secondary degraded capillaries is not
accounted in NP
and NA
i;j
. Therefore, this approach allows to reproduce a spontaneous
i;j
nature of capilliary degradation. Then, the microcirculatory network model provides new
data on the surface density of normal and angiogenic capillaries, PC and AC, and the
density of blood flow Q:
PC
AC
0
B
X
B
¼
B
i;j
@
k
;
norm
r
i1<rk<ri
0
B
X
B
¼
B
i;j
@
k
;
ang
r
i1<rk<ri
pdkl
pdkl
k;i
k;i
1
C
C
C
A
1
C
C
C
A
4
3
r
pr
=
=
i
3
4
3
r
pr
i
3
3
i1
3
i1
;
;
(8.14)
where k
norm
and k
branch off from arterioles (as discussed in Section 8.3.3, we assume that all oxygen
goes into the tissue through them), d
capillary, l
is the length of the part of the k-th capillary located in the i-th layer. The
k;i
resulting data arrays are then interpolated by cubic splines, i.e., piecewise smooth cubic
polynomials PC
(see Ref. [398] for algorithm of cubic spline interpolation).
0
B
X
B
¼
Q
B
i;j
@
k
;
first
r
i1<rk<ri
are normal and angiogenic capillaries, k
ang
k
ðrÞ, ACjðrÞ, QjðrÞ, which are then used in the model of tumor growth
j
1
3
i1
;
are capillaries that
first
C
l
k;i
q
k
l
4
C
=
C
A
k
3
pr
r
i
3
and lkare the diameter a nd length of the k-th
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Using the obtained information, the inflows of oxygen and glucose Q
and QSare
O
2
calculated in the tumor growth model. The local inflow of glucose is proportional to the
diffusive permeability of capillaries, which is different for normal and angiogenic
capillaries, and also to the difference in its concentrations in blood and in tissue. The
inflow of oxygen is calculated under the assumptions that (1) oxygen transvascular
transport is fast enough for values of oxygen pressure in capillary and in tissue to become
equal along one capillary length (which has already been discussed) and (2) the rate of
binding to and unbinding from hemoglobin is sufficiently fast for oxygen so that use of
quasi-stationary approximation is legitimate. Together with more general model
assumption that microvasculature volume is negligible compared with tissue volume, this
approach results in term for oxygen inflow proportional to the difference between
hemoglobin saturation under two values of unbound oxygen concentrationdthe one in
arterial blood, which enters the capillaries, and the one in tissue. The form of the function
of oxyhemoglobin fraction depending on oxygen pressure, or oxygenehemoglobin
dissociation curve, is well known for already about a century [399]. Thus, the inflow of
nutrients for the time interval ½t
Q
ðr; tÞ¼½PPCPCjðrÞþPACACjðrÞ½S
S
Q
ðr; tÞ¼Q
O
2
HSðO
ðr; tÞÞ ¼ O2ðr; tÞ
2
; t
j
jþ1
norm
QjðrÞHSO
O
2
is calculated according to the following equations:
Sðr; tÞ;
blood
art
HSðO
2
2
!
hill
f
O
=
4
=
2
2
1 þ O
ðr; tÞÞ;
ðr; tÞ
2
3
!
hill
f
O
=
5
2
;
(8.15)
where P
glucose; S
and PACare the permeability values of normal and angiogenic capillaries for
PC
is the level of glucose in blood; Q
blood
norm
is the normalization constant for the
O
2
oxygen inflow selected so that, in the absence of tumor cells, the oxygen level is equal to
its predetermined normal physiological level in tissue; O
artery;fO
HSðO
and hill are coefficients that fit the oxygenehemoglobin dissociation curve
2
Þ.
2
art
is oxygen concentration in
2
The resulting inflows of nutrients are substituted into the equations of tumor growth
submodel. These equations include terms describing nutrients inflow, diffusion and
consumption by cells. Altogether this defines nutrients dynamics. Next, the tumor growth
model is simulated until the next moment of the models conjugation is reached. After that,
the above algorithm is repeated.
At the initial moment of time t ¼ 0 in the submodel of tumor growth, the distributions of
surface densities of capillaries PC
ðrÞ and AC0ðrÞ are determined on the basis of the
0
normal microcirculatory network according to the algorithm described above. Other initial
conditions correspond to normal tissue with a small population of tumor cells in the center
of the computational domain.
Hemodynamics in normal and angiogenic capillary networks 185
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proliferating tumor cells
all tumor cells
tumor with necrosis
2.5
(A)
2.0
1.5
1.0
0.5
0.0
0 1.5 3 4.5 6
2.5
(B)
2.0
1.5
1.0
0.5
0.0
01.534.56
day 5
mm mm
normal capillaries
all capillaries
day 1
2.5
(C)
2.0
1.5
1.0
0.5
0.0
0 1.5 3 4.5 6
glucose
oxygen
VEGF
mm
day 8
Figure 8.6
Distribution of variables of coupled model of tumor growth in tissue that account for
microcirculatory network remodeling, during several days of tumor growth.
Fig. 8.6 demonstrates examples of distributions of model variables during first days of
tumor growth. It is seen that the resulting model is able to adequately reproduce known
physiological features at the beginning of tumor growth. Capillaries degrade inside the
tumor core, leading to reduction in inflow of nutrients and subsequent formation of
necrosis, while proliferating tumor cells are located at the tumor rim. Angiogenic
capillaries are formed mainly in the peritumoral region, providing an increased supply of
metabolites in this area, noticed by a slight increase in oxygen concentration there.
Unfortunately, the practical implementation of the presented model meets restrictions,
already discussed in Section 8.4.1, i.e., enormous computational costs. Utilizing such
approach for the investigation of tumor growth larger than 1 cm in diameter obviously
requires utilization of a supercomputer. Obtaining qualitative physiologically meaningful
results requires a considerable variation of model parameters. This also increases the
computational burden.
186 Chapter 8
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Conclusions
In this study, a novel modeling approach was presented, which allows to generate
microcirculatory networks, possessing similar geometric and functional characteristics to
the ones of a real microcirculatory network. In program implementation of the method, the
generation of the network structure obeys a number of requirements. One requirement is
matching the distributions of microvessels’ lengths and diameters obtained in experimental
work on the microcirculatory network of breast cancer [339]. Blood flow calculations in
the generated microcirculatory networks are based on the Poiseuille law with nonlinear
conductivity term and the law of mass conservation. The adequacy of the model
prediction, i.e., the ability of the generated microcirculatory network to virtually supply all
cells in tissue with key metabolites, was checked by statistical analyses. This analysis
confirmed the space-uniformity of distributions of capillaries surface area and blood flow
in capillaries emanating from the arterioles. The homogeneous surface area distribution is
crucial for glucose and most other nutrients supply, while the uniform blood flow is
important for oxygen delivery. The investigations of the model showed that, as the number
of microvessels in the network increases, the total blood flow at first increases and then
approaches the saturation threshold, what is in a good agreement with common
physiological sense. This implies that increase in oxygen supply is always less than
corresponding increase in microvascular density, which should be accounted for in
experiments. This result served as the basis for the introduction of a new physiologically
motivated term of oxygen inflow in the model of tumor growth. We may conclude that the
oxygen supply is flow limited rather than diffusion limited, overcoming a common fallacy
in the models of tumor growth (see, e.g., Refs. [400,401]). This already led to a separate
fruitful investigation [380], the results of which are mentioned further. An attempt for
complete coupling of microvasculature and tumor models was performed and resulted in a
working and physiologically adequate prototype. However, its further development is
constrained by serious computational limitations.
A significant result obtained here is the model prediction that the normalization of
structure of tumor microvessels and the concomitant decrease in tumor-associated edema
as a result of antiangiogenic therapy should lead to less than twofold local increase in
tumor perfusion within the range of standard observed values of tumor interstitial pressure
before therapeutic intervention. This result agrees well with the experimental data
[386,503]. It should be emphasized that the experimental measurements, made together
with the measurements of tumor perfusion in the work [503], indicate a simultaneous
almost fourfold increase in the oxygen pressure within the tumor, whereas two- to
threefold growths in intratumoral oxygen pressure in different corresponding experiments
have been reported in work [383]. These facts lead to doubts that an increase in tumor
blood flow is the onlydor even the maindcause for transient alleviation of tumor
Hemodynamics in normal and angiogenic capillary networks 187
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hypoxia. In the work [380], it was demonstrated that this phenomenon may also result
from changes in tumor metabolism and in the rates of inflow of key nutrients in the tissue,
caused by the therapy. Thus, the results of the present study support this hypothesis.
Of course, the real tumor angiogenic capillary network differs from the simulated one in
that it does not constitute a tree structure, since the sprouting angiogenic capillaries
connect with each other and with the existing capillaries rather chaotically; moreover,
tumor angiogenesis results in a large number of blind ends of capillaries. Nevertheless,
preliminary estimates show that deviations from the tree structure (i.e., adding bridges of
capillaries) do not affect the obtained result, and further research is planned in this
direction.
CHAPTER 9
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Computational hemodynamics in vascular
surgery
9.1 Introduction
In this chapter, we address several clinical applications of regional patient-specific
hemodynamic models. Clinical practice imposes rather demanding requirements on a
mathematical model and numerical methods used for predictive simulations. An ideal
model must take into account all patients’ data available in routinely clinical diagnosis and
may not require additional costly or time-consuming measurements. The model should
minimize the number of parameters to be identified, and the remaining parameters should
have clear physiological meaning. It should be feasible to run simulations in a
conventional clinical environment, which means only a few minutes or even seconds of
simulation time on a personal computer or a workstation. The model should produce a
result demanded by clinicians, which often comes down to one or few statistics related to
several scenarios of an endovascular intervention.
The one-dimensional hemodynamic model of regional blood circulation introduced in
Chapter 7 meets these requirements to a large extent. Although the considered clinical
applications address different regional vasculatures, including leg arteries [242,253],
coronary arteries [17,260,264,320,402e406], and cerebral arteries [281,314,407,408], the
corresponding regional hemodynamic models have many common features: they are based
on the same methods of 3D image segmentation, they account for autoregulation, they are
not closed and use inlet boundary condition from the heart or the aorta, they use
aggregated or virtual venous system, and they use the same method of parameters
identification and microcirculation resistance.
9.2 Identification of parameters
Clinical applications of hemodynamic models require personalized simulations. The
geometry of the vascular bed can be recovered by methods discussed in Chapter 3. Apart
from the geometry, hemodynamic models operate with other functional parameters that
have to be patient specific such as stroke volume, peripheral resistance, and vessel wall
stiffness. For some applications, systolic/diastolic pressure and flow rate waveforms (if
available) may help to fit better the model to the patient circulation. General information
Personalized Computational Hemodynamics. https://doi.org/10.1016/B978-0-12-815653-7.00009-9
Copyright © 2020 Elsevier Inc. All rights reserved.
189
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such as height, weight, age, smoking status, and clinical records can be accounted in
model parameters as well.
The stroke volume is the integral of flow rate at the entrance to the aorta within one cardiac
cycle. If known, the flow rate profile can be used in patient-specific simulations. Usually,
such information is not available in a clinic due to the extreme precision required by such
measurement. By default, the averaged physiological flow rate profile Q
ðtÞ (Fig. 9.1)is
heart
scaled in accordance with the measured stroke volume and the heart rate. For any patient,
the heart rate at rest is available, and the argument t of function Q
ðtÞ can be scaled to
heart
match the cardiac cycle. The magnitude of the stroke volume can also be adjusted by
multiplying the function Q
corresponds to a stroke volume of 62 mL. In the absence of data on the stroke volume, a
ðtÞ by a weight factor: ainQ
heart
ðtÞ. The value ain¼ 1
heart
in
is selected to fit the simulated and measured blood flow rates at the points close to the aortic
arch. By default, the central venous pressure p
is assumed to be equal to 8 mm Hg. This
v
value can be changed to patient’s pressure if available.
For each terminal artery, it is necessary to select the hydraulic resistance R of the regional
microcirculation. The initial values of R are chosen so that the simulated velocities and
pressures correspond to the physiological norm [239,240]. For instance, for
brachiocephalic arteries (Fig. 9.2), the following resistances result in blood flow rates close
to physiological flow rates: for the aortic arch passing into the thoracic aorta
R ¼ 250 dyn s/cm
5
, for the subclavian arteries (LSA and RSA) R ¼ 4 kdyn s/cm5, for the
Scaling of the heart outflow condition. Four curves correspond to different heart rates and to the
same stroke volume 62 mL: 60 bpm (beats per minute) (black), 80 bpm (red [dark gray in print
version]), 100 bpm (blue [gray in print version]), and 120 bpm (pink [light gray in print
Figure 9.1
version]).
Computational hemodynamics in vascular surgery 191
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Figure 9.2
3D and 1D structures of the vasculature. Points indicate positions of ultrasound measurements
of blood flow velocity. The designations: right (R), left (L), carotid artery (CA), common carotid
artery (CCA), internal carotid artery (ICA), external carotid artery (ECA), vertebral artery (VA),
subclavian artery (SA).
external carotid arteries (left external carotid artery and posterior external carotid artery)
R ¼ 40 kdyn s/cm
5
, and for remaining terminal arteries R ¼ 400 kdyn s/cm5.
When modeling the blood flow of a particular patient, the resistances need to be changed
to fit the measured blood flow velocity. For instance, in case of brachiocephalic arteries,
the measurement points are shown in the right picture of Fig. 9.2. With a significant
deviation (more than 20%) of the measured velocity from the calculated one, the
resistance of microvasculature lying downstream of the measurement site has to be
corrected as follows: in the case of overestimated velocities, R should be increased and
vice versa. The increment should not exceed 10% of the original resistance value R.
Larger deviations may be out of physiological range. If the velocities do not match at
several points of measurement, the resistance should be modified first in the terminal
vessels located closer to the aortic arch. In all considered cases, such procedure allows to
achieve deviation of measured velocities from calculated ones with the error less than
20%. To fit the model better, one needs to adjust wall elasticity (stiffness) parameters.
The vessel wall stiffness can be interpreted as the velocity of small disturbance
propagation c
(pulse wave propagation) along the vessel k under zero transmural pressure
k
(7.47). Higher velocity corresponds to a relatively stiff wall, and lower velocity indicates
more flexible and compliant wall. The pulse wave velocity is related to vessel compliance
[409] and elastic modulus [410]. This velocity can be measured in arteries directly
[411,412]; however, necessary equipment may not be available in a hospital and,
moreover, the measurement gives only averaged characteristics for the whole body.
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On the other hand, the pulse wave velocity in different arteries is available in the literature
[413,414]. These data can be adapted to physiological status and lifestyle conditions of the
patient. Such approach gives a good initial approximation of the pulse wave velocity if we
are given basic characteristics of the patient and his or her clinical record. Further
refinement of the parameter c
for each individual vessel is possible if ultrasound
k
measurement is available in it: if the simulated velocity deviates from the measured one
by more than 5%, then c
in the selected vessel k is adjusted, whereas in the neighboring
k
vessels, the parameter is adjusted only partly (one half of the adjustment). Pulse wave
velocities may vary considerably from one vessel to another one, so the adjustment may
change the velocity up to 50% of the initial value, which is within the range of c
k
variability as shown in the following.
Morphological changes due to aging result in vessel stiffness changes: arteries become less
compliant and more stiff with age. The simplest formula accounting for this phenomenon
is given by:
8
>
>
; y < y
c
>
c
0
ðyÞ¼
a
>
>
<
caþðy yaÞ
>
>
>
>
>
cb; y > y
:
cb c
yb y
a
; ya y y
a
a
b
b
(9.1)
where y is the age of a patient and c
(y
¼ 20 years) and an old man (yb¼ 80 years), respectively. Clinical studies [415,416]
a
provide the parameters c
¼ 500 cm/s and cb¼ 650 cm/s.
a
and cbare the pulse wave velocity for a young man
a
We note that Eq. (9.1) may need parameter adjustment depending of patient’s gender. For
instance, according to Fig. 9.3, the left and right coronary arteries of women lose much of
flexibility 10 years later than for men (compare groups V and VI).
Apart from age, the pulse wave velocity is affected by other factors, which can be
accounted by formula.
c ¼c
Factor a
introduces dependence on patient’s blood pressure [417]. It is set to 1 for the
hyper
normal systolic pressure p
with strong hypertension p
where p
Factor a
of factor a
is patient’s systolic pressure.
syst
accounts vessel stiffening for smokers [418]. Table 9.1 presents dependence
smoke
on smoking status.
smoke
a
0
hyperasmoke
¼ 120 mm Hg and increases linearly to 1.5 for patients
norm
¼ 220 mm Hg:
hyper
¼1 þ
a
hyper
2p
p
hyper
syst
a
p
p
: (9.2)
sport
norm
; (9.3)
norm
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